Twin primes and the $3$-sphere

TL;DR

This paper explores the relationship between twin primes and the 3-sphere, proposing a series whose convergence is equivalent to the twin prime conjecture, thus linking prime distribution to geometric and polynomial structures.

Contribution

It introduces a novel series connected to twin primes and the 3-sphere, providing a new approach to the twin prime conjecture through geometric and algebraic methods.

Findings

Series converges iff finitely many twin primes exist

Connections established between group points on the 3-sphere and known groups

Links made between Chebyshev polynomials and prime pairs

Abstract

We investigate the group of points of the $3$-sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many pairs of twin primes. Hence to prove that that the series diverges is to prove the twin prime conjecture.